HOME

TheInfoList



OR:

A covering of a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
X is a
continuous map In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
\pi : E \rightarrow X with special properties.


Definition

Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a
discrete space In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are ''isolated'' from each other in a certain sense. The discrete topology is the finest top ...
D and for every x \in X an open neighborhood U \subset X, such that \pi^(U)= \displaystyle \bigsqcup_ V_d and \pi, _:V_d \rightarrow U is a
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isom ...
for every d \in D . Often, the notion of a covering is used for the covering space E as well as for the map \pi : E \rightarrow X. The open sets V_ are called sheets, which are uniquely determined up to a homeomorphism if U is
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
. For each x \in X the discrete subset \pi^(x) is called the fiber of x. The degree of a covering is the
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
of the space D. If E is
path-connected In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties ...
, then the covering \pi : E \rightarrow X is denoted as a path-connected covering.


Examples

* For every topological space X there exists the covering \pi:X \rightarrow X with \pi(x)=x, which is denoted as the trivial covering of X. * The map r \colon \mathbb \to S^1 with r(t)=(\cos(2 \pi t), \sin(2 \pi t)) is a covering of the
unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
S^1. The base of the covering is S^1 and the covering space is \mathbb. For any point x = (x_1, x_2) \in S^1 such that x_1 > 0, the set U := \ is an open neighborhood of x. The preimage of U under r is ::r^(U)=\displaystyle\bigsqcup_ \left( n - \frac 1 4, n + \frac 1 4\right) :and the sheets of the covering are V_n = (n - 1/4, n+1/4) for n \in \mathbb. The fiber of x is ::r^(x) = \. * Another covering of the unit circle is the map q \colon S^1 \to S^1 with q(z)=z^ for some n \in \mathbb. For an open neighborhood U of an x \in S^1, one has: ::q^(U)=\displaystyle\bigsqcup_^ U. * A map which is a
local homeomorphism In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure. If f : X \to Y is a local homeomorphism, X is said to be an � ...
but not a covering of the unit circle is p \colon \mathbb \to S^1 with p(t)=(\cos(2 \pi t), \sin(2 \pi t)). There is a sheet of an open neighborhood of (1,0), which is not mapped homeomorphically onto U.


Properties


Local homeomorphism

Since a covering \pi:E \rightarrow X maps each of the disjoint open sets of \pi^(U) homeomorphically onto U it is a local homeomorphism, i.e. \pi is a continuous map and for every e \in E there exists an open neighborhood V \subset E of e, such that \pi, _V : V \rightarrow \pi(V) is a homeomorphism. It follows that the covering space E and the base space X locally share the same properties. * If X is a connected and non-orientable manifold, then there is a covering \pi:\tilde X \rightarrow X of degree 2, whereby \tilde X is a connected and orientable manifold. * If X is a connected
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
, then there is a covering \pi:\tilde X \rightarrow X which is also a Lie group homomorphism and \tilde X := \ is a Lie group. * If X is a
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
, then it follows for a covering \pi:E \rightarrow X that E is also a graph. * If X is a connected
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
, then there is a covering \pi:\tilde X \rightarrow X, whereby \tilde X is a connected and
simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spa ...
manifold. * If X is a connected
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
, then there is a covering \pi:\tilde X \rightarrow X which is also a holomorphic map and \tilde X is a connected and simply connected Riemann surface.


Factorisation

Let p,q and r be continuous maps, such that the diagram commutes. * If p and q are coverings, so is r. * If p and r are coverings, so is q.


Product of coverings

Let X and X' be topological spaces and p:E \rightarrow X and p':E' \rightarrow X' be coverings, then p \times p':E \times E' \rightarrow X \times X' with (p \times p')(e, e') = (p(e), p'(e')) is a covering.


Equivalence of coverings

Let X be a topological space and p:E \rightarrow X and p':E' \rightarrow X be coverings. Both coverings are called equivalent, if there exists a homeomorphism h:E \rightarrow E', such that the diagram commutes. If such a homeomorphism exists, then one calls the covering spaces E and E'
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
.


Lifting property

An important property of the covering is, that it satisfies the
lifting property In mathematics, in particular in category theory, the lifting property is a property of a pair of morphisms in a category. It is used in homotopy theory within algebraic topology to define properties of morphisms starting from an explicitly give ...
, i.e.: Let I be the
unit interval In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis ...
and p:E \rightarrow X be a covering. Let F:Y \times I \rightarrow X be a continuous map and \tilde F_0:Y \times \ \rightarrow E be a lift of F, _, i.e. a continuous map such that p \circ \tilde F_0 = F, _. Then there is a uniquely determined, continuous map \tilde F:Y \times I \rightarrow E, which is a lift of F, i.e. p \circ \tilde F = F. If X is a path-connected space, then for Y=\ it follows that the map \tilde F is a lift of a path in X and for Y=I it is a lift of a
homotopy In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deform ...
of paths in X. Because of that property one can show, that the
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, o ...
\pi_(S^1) of the unit circle is an
infinite cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative binar ...
, which is generated by the homotopy classes of the loop \gamma: I \rightarrow S^1 with \gamma (t) = (\cos(2 \pi t), \sin(2 \pi t)). Let X be a path-connected space and p:E \rightarrow X be a connected covering. Let x,y \in X be any two points, which are connected by a path \gamma, i.e. \gamma(0)= x and \gamma(1)= y. Let \tilde \gamma be the unique lift of \gamma, then the map : L_:p^(x) \rightarrow p^(y) with L_(\tilde \gamma (0))=\tilde \gamma (1) is
bijective In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
. If X is a path-connected space and p: E \rightarrow X a connected covering, then the induced
group homomorphism In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) w ...
: p_: \pi_(E) \rightarrow \pi_(X) with p_(
gamma Gamma (uppercase , lowercase ; ''gámma'') is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter r ...
= \circ \gamma/math>, is
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...
and the
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
p_(\pi_1(E)) of \pi_1(X) consists of the homotopy classes of loops in X, whose lifts are loops in E.


Branched covering


Definitions


Holomorphic maps between Riemann surfaces

Let X and Y be
Riemann surfaces In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
, i.e. one dimensional
complex manifolds In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a c ...
, and let f: X \rightarrow Y be a continuous map. f is holomorphic in a point x \in X, if for any
charts A chart (sometimes known as a graph) is a graphical representation for data visualization, in which "the data is represented by symbols, such as bars in a bar chart, lines in a line chart, or slices in a pie chart". A chart can represent tabu ...
\phi _x:U_1 \rightarrow V_1 of x and \phi_:U_2 \rightarrow V_2 of f(x), with \phi_x(U_1) \subset U_2, the map \phi _ \circ f \circ \phi^ _x: \mathbb \rightarrow \mathbb is holomorphic. If f is holomorphic at all x \in X, we say f is holomorphic. The map F =\phi _ \circ f \circ \phi^ _x is called the local expression of f in x \in X. If f: X \rightarrow Y is a non-constant, holomorphic map between compact Riemann surfaces, then f is
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element o ...
and an
open map In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y. Likewise, ...
, i.e. for every open set U \subset X the
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
f(U) \subset Y is also open.


Ramification point and branch point

Let f: X \rightarrow Y be a non-constant, holomorphic map between compact Riemann surfaces. For every x \in X there exist charts for x and f(x) and there exists a uniquely determined k_x \in \mathbb, such that the local expression F of f in x is of the form z \mapsto z^. The number k_x is called the ramification index of f in x and the point x \in X is called a ramification point if k_x \geq 2. If k_x =1 for an x \in X, then x is unramified. The image point y=f(x) \in Y of a ramification point is called a branch point.


Degree of a holomorphic map

Let f: X \rightarrow Y be a non-constant, holomorphic map between compact Riemann surfaces. The degree \operatorname(f) of f is the cardinality of the fiber of an unramified point y=f(x) \in Y, i.e. \operatorname(f):=, f^(y), . This number is well-defined, since for every y \in Y the fiber f^(y) is discrete and for any two unramified points y_1,y_2 \in Y, it is: , f^(y_1), =, f^(y_2), . It can be calculated by: : \sum_ k_x = \operatorname(f)


Branched covering


Definition

A continuous map f: X \rightarrow Y is called a branched covering, if there exists a
closed set In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a ...
with dense complement E \subset Y, such that f_:X \smallsetminus f^(E) \rightarrow Y \smallsetminus E is a covering.


Examples

* Let n \in \mathbb and n \geq 2, then f:\mathbb \rightarrow \mathbb with f(z)=z^n is branched covering of degree n, whereby z=0 is a branch point. * Every non-constant, holomorphic map between compact Riemann surfaces f: X \rightarrow Y of degree d is a branched covering of degree d.


Universal covering


Definition

Let p: \tilde X \rightarrow X be a
simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spa ...
covering. If \beta : E \rightarrow X is another simply connected covering, then there exists a uniquely determined homeomorphism \alpha : \tilde X \rightarrow E, such that the diagram commutes. This means that p is, up to equivalence, uniquely determined and because of that
universal property In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fr ...
denoted as the universal covering of the space X.


Existence

A universal covering does not always exist, but the following properties guarantee its existence: Let X be a connected,
locally simply connected In mathematics, a locally simply connected space is a topological space that admits a basis of simply connected sets. Every locally simply connected space is also locally path-connected and locally connected. The circle is an example of a locally ...
topological space; then, there exists a universal covering p:\tilde X \rightarrow X. \tilde X is defined as \tilde X := \/\text and p:\tilde X \rightarrow X by p(
gamma Gamma (uppercase , lowercase ; ''gámma'') is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter r ...
:=\gamma(1). The
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
on \tilde X is constructed as follows: Let \gamma:I \rightarrow X be a path with \gamma(0)=x_0. Let U be a simply connected neighborhood of the endpoint x=\gamma(1), then for every y \in U the paths \sigma_y inside U from x to y are uniquely determined up to
homotopy In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deform ...
. Now consider \tilde U:=\/\text, then p_: \tilde U \rightarrow U with p( gamma.\sigma_y=\gamma.\sigma_y(1)=y is a bijection and \tilde U can be equipped with the final topology of p_. The fundamental group \pi_(X,x_0) = \Gamma acts freely through (
gamma Gamma (uppercase , lowercase ; ''gámma'') is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter r ...
tilde x \mapsto gamma.\tilde x/math> on \tilde X and \psi:\Gamma \backslash \tilde X \rightarrow X with \psi( Gamma \tilde x=\tilde x(1) is a homeomorphism, i.e. \Gamma \backslash \tilde X \cong X .


Examples

* r \colon \mathbb \to S^1 with r(t)=(\cos(2 \pi t), \sin(2 \pi t)) is the universal covering of the unit circle S^1. * p \colon S^n \to \mathbbP^n \cong \\backslash S^n with p(x)= /math> is the universal covering of the
projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
\mathbbP^n for n>1. * q \colon SU(n) \ltimes \mathbb \to U(n) with q(A,t)= \begin \exp(2 \pi i t) & 0\\ 0 & I_ \end_\vphantom A is the universal covering of the
unitary group In mathematics, the unitary group of degree ''n'', denoted U(''n''), is the group of unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group . Hyperorthogonal group is ...
U(n). * Since SU(2) \cong S^3, it follows that the
quotient map In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient to ...
f:SU(2) \rightarrow \mathbb \backslash SU(2) \cong SO(3) is the universal covering of the SO(3). * A topological space which has no universal covering is the
Hawaiian earring In mathematics, the Hawaiian earring \mathbb is the topological space defined by the union of circles in the Euclidean plane \R^2 with center \left(\tfrac,0\right) and radius \tfrac for n = 1, 2, 3, \ldots endowed with the subspace topology: ...
: X = \bigcup_\left\ One can show that no neighborhood of the origin (0,0) is simply connected.


Deck transformation


Definition

Let p:E \rightarrow X be a covering. A deck transformation is a homeomorphism d:E \rightarrow E, such that the diagram of continuous maps commutes. Together with the composition of maps, the set of deck transformation forms a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
\operatorname(p), which is the same as \operatorname(p).


Examples

* Let q \colon S^1 \to S^1 be the covering q(z)=z^ for some n \in \mathbb , then the map d:S^1 \rightarrow S^1 : z \mapsto z \, e^ is a deck transformation and \operatorname(q)\cong \mathbb/ \mathbb. * Let r \colon \mathbb \to S^1 be the covering r(t)=(\cos(2 \pi t), \sin(2 \pi t)), then the map d_k:\mathbb \rightarrow \mathbb : t \mapsto t + k with k \in \mathbb is a deck transformation and \operatorname(r)\cong \mathbb.


Properties

Let X be a path-connected space and p:E \rightarrow X be a connected covering. Since a deck transformation d:E \rightarrow E is
bijective In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
, it permutes the elements of a fiber p^(x) with x \in X and is uniquely determined by where it sends a single point. In particular, only the identity map fixes a point in the fiber. Because of this property every deck transformation defines a
group action In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
on E, i.e. let U \subset X be an open neighborhood of a x \in X and \tilde U \subset E an open neighborhood of an e \in p^(x), then \operatorname(p) \times E \rightarrow E: (d,\tilde U)\mapsto d(\tilde U) is a
group action In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
.


Normal coverings


Definition

A covering p:E \rightarrow X is called normal, if \operatorname(p) \backslash E \cong X. This means, that for every x \in X and any two e_0,e_1 \in p^(x) there exists a deck transformation d:E \rightarrow E, such that d(e_0)=e_1.


Properties

Let X be a path-connected space and p:E \rightarrow X be a connected covering. Let H=p_(\pi_1(E)) be a
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
of \pi_1(X), then p is a normal covering iff H is a
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
of \pi_1(X). If p:E \rightarrow X is a normal covering and H=p_(\pi_1(E)), then \operatorname(p) \cong \pi_1(X)/H. If p:E \rightarrow X is a path-connected covering and H=p_(\pi_1(E)), then \operatorname(p) \cong N(H)/H, whereby N(H) is the normaliser of H. Let E be a topological space. A group \Gamma acts discontinuously on E, if every e \in E has an open neighborhood V \subset E with V \neq \empty, such that for every \gamma \in \Gamma with \gamma V \cap V \neq \empty one has d_1 = d_2. If a group \Gamma acts discontinuously on a topological space E, then the
quotient map In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient to ...
q: E \rightarrow \Gamma \backslash E with q(e)=\Gamma e is a normal covering. Hereby \Gamma \backslash E = \ is the quotient space and \Gamma e = \ is the
orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as ...
of the group action.


Examples

* The covering q \colon S^1 \to S^1 with q(z)=z^ is a normal coverings for every n \in \mathbb. * Every simply connected covering is a normal covering.


Calculation

Let \Gamma be a group, which acts discontinuously on a topological space E and let q: E \rightarrow \Gamma \backslash E be the normal covering. * If E is path-connected, then \operatorname(q) \cong \Gamma. * If E is simply connected, then \operatorname(q)\cong \pi_1(X).


Examples

* Let n \in \mathbb. The antipodal map g:S^n \rightarrow S^n with g(x)=-x generates, together with the composition of maps, a group D(g) \cong \mathbb and induces a group action D(g) \times S^n \rightarrow S^n, (g,x)\mapsto g(x), which acts discontinuously on S^n. Because of \mathbb \backslash S^n \cong \mathbbP^n it follows, that the quotient map q \colon S^n \rightarrow \mathbb\backslash S^n \cong \mathbbP^n is a normal covering and for n > 1 a universal covering, hence \operatorname(q)\cong \mathbb\cong \pi_1() for n > 1. * Let SO(3) be the
special orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
, then the map f:SU(2) \rightarrow SO(3) \cong \mathbb \backslash SU(2) is a normal covering and because of SU(2) \cong S^3, it is the universal covering, hence \operatorname(f) \cong \mathbb \cong \pi_1(SO(3)). * With the group action (z_1,z_2)*(x,y)=(z_1+(-1)^x,z_2+y) of \mathbb on \mathbb, whereby (\mathbb,*) is the
semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in wh ...
\mathbb \rtimes \mathbb , one gets the universal covering f: \mathbb \rightarrow (\mathbb \rtimes \mathbb) \backslash \mathbb \cong K of the
klein bottle In topology, a branch of mathematics, the Klein bottle () is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a ...
K, hence \operatorname(f) \cong \mathbb \rtimes \mathbb \cong \pi_1(K). * Let T = S^1 \times S^1 be the
torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does n ...
which is embedded in the \mathbb. Then one gets a homeomorphism \alpha: T \rightarrow T: (e^,e^) \mapsto (e^,e^), which induces a discontinuous group action G_ \times T \rightarrow T, whereby G_ \cong \mathbb. It follows, that the map f: T \rightarrow G_ \backslash T \cong K is a normal covering of the klein bottle, hence \operatorname(f) \cong \mathbb. * Let S^3 be embedded in the \mathbb. Since the group action S^3 \times \mathbb \rightarrow S^3: ((z_1,z_2), \mapsto (e^z_1,e^z_2) is discontinuously, whereby p,q \in \mathbb are
coprime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
, the map f:S^3 \rightarrow \mathbb \backslash S^3 =: L_ is the universal covering of the lens space L_, hence \operatorname(f) \cong \mathbb \cong \pi_1(L_).


Galois correspondence

Let X be a connected and
locally simply connected In mathematics, a locally simply connected space is a topological space that admits a basis of simply connected sets. Every locally simply connected space is also locally path-connected and locally connected. The circle is an example of a locally ...
space, then for every
subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
H\subseteq \pi_1(X) there exists a path-connected covering \alpha:X_H \rightarrow X with \alpha_(\pi_1(X_H))=H. Let p_1:E \rightarrow X and p_2: E' \rightarrow X be two path-connected coverings, then they are equivalent iff the subgroups H = p_(\pi_1(E)) and H'=p_(\pi_1(E')) are conjugate to each other. Let X be a connected and locally simply connected space, then, up to equivalence between coverings, there is a bijection: \begin \qquad \displaystyle \ & \longleftrightarrow & \displaystyle \ \\ H & \longrightarrow & \alpha:X_H \rightarrow X \\ p_\#(\pi_1(E))&\longleftarrow & p \\ \displaystyle \ & \longleftrightarrow & \displaystyle \ \\ H & \longrightarrow & \alpha:X_H \rightarrow X \\ p_\#(\pi_1(E))&\longleftarrow & p \end For a sequence of subgroups \displaystyle \ \subset H \subset G \subset \pi_1(X) one gets a sequence of coverings \tilde X \longrightarrow X_H \cong H \backslash \tilde X \longrightarrow X_G \cong G \backslash \tilde X \longrightarrow X\cong \pi_1(X) \backslash \tilde X . For a subgroup H \subset \pi_1(X) with
index Index (or its plural form indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on a Halo megastru ...
\displaystyle pi_1(X):H= d , the covering \alpha:X_H \rightarrow X has degree d.


Classification


Definitions


Category of coverings

Let X be a topological space. The objects of the
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...
\boldsymbol are the coverings p:E \rightarrow X of X and the
morphisms In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
between two coverings p:E \rightarrow X and q:F\rightarrow X are continuous maps f:E \rightarrow F, such that the diagram commutes.


G-Set

Let G be a
topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
. The
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...
\boldsymbol is the category of sets which are G-sets. The morphisms are G-maps \phi:X \rightarrow Y between G-sets. They satisfy the condition \phi(gx)=g \, \phi(x) for every g \in G.


Equivalence

Let X be a connected and locally simply connected space, x \in X and G = \pi_1(X,x) be the fundamental group of X. Since G defines, by lifting of paths and evaluating at the endpoint of the lift, a group action on the fiber of a covering, the
functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
F:\boldsymbol \longrightarrow \boldsymbol: p \mapsto p^(x) is an
equivalence of categories In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences fr ...
.


Applications

An important practical application of covering spaces occurs in charts on SO(3), the rotation group. This group occurs widely in engineering, due to 3-dimensional rotations being heavily used in
navigation Navigation is a field of study that focuses on the process of monitoring and controlling the movement of a craft or vehicle from one place to another.Bowditch, 2003:799. The field of navigation includes four general categories: land navigation ...
, nautical engineering, and
aerospace engineering Aerospace engineering is the primary field of engineering concerned with the development of aircraft and spacecraft. It has two major and overlapping branches: aeronautical engineering and astronautical engineering. Avionics engineering is s ...
, among many other uses. Topologically, SO(3) is the
real projective space In mathematics, real projective space, denoted or is the topological space of lines passing through the origin 0 in It is a compact, smooth manifold of dimension , and is a special case of a Grassmannian space. Basic properties Construction A ...
RP3, with fundamental group Z/2, and only (non-trivial) covering space the hypersphere ''S''3, which is the group Spin(3), and represented by the unit
quaternions In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...
. Thus quaternions are a preferred method for representing spatial rotations – see
quaternions and spatial rotation Unit quaternions, known as ''versors'', provide a convenient mathematical notation for representing spatial orientations and rotations of elements in three dimensional space. Specifically, they encode information about an axis-angle rotation abou ...
. However, it is often desirable to represent rotations by a set of three numbers, known as
Euler angles The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body with respect to a fixed coordinate system.Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 189–207 (E478PDF/ref> Th ...
(in numerous variants), both because this is conceptually simpler for someone familiar with planar rotation, and because one can build a combination of three
gimbal A gimbal is a pivoted support that permits rotation of an object about an axis. A set of three gimbals, one mounted on the other with orthogonal pivot axes, may be used to allow an object mounted on the innermost gimbal to remain independent of ...
s to produce rotations in three dimensions. Topologically this corresponds to a map from the 3-torus ''T''3 of three angles to the real projective space RP3 of rotations, and the resulting map has imperfections due to this map being unable to be a covering map. Specifically, the failure of the map to be a local homeomorphism at certain points is referred to as
gimbal lock Gimbal lock is the loss of one degree of freedom in a three-dimensional, three-gimbal mechanism that occurs when the axes of two of the three gimbals are driven into a parallel configuration, "locking" the system into rotation in a degenerate t ...
, and is demonstrated in the animation at the right – at some points (when the axes are coplanar) the
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * ...
of the map is 2, rather than 3, meaning that only 2 dimensions of rotations can be realized from that point by changing the angles. This causes problems in applications, and is formalized by the notion of a covering space.


See also

*
Bethe lattice In statistical mechanics and mathematics, the Bethe lattice (also called a regular tree) is an infinite connected cycle-free graph where all vertices have the same number of neighbors. The Bethe lattice was introduced into the physics literature ...
is the universal cover of a
Cayley graph In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cay ...
*
Covering graph In the mathematical discipline of graph theory, a graph is a covering graph of another graph if there is a covering map from the vertex set of to the vertex set of . A covering map is a surjection and a local isomorphism: the neighbourhood of ...
, a covering space for an
undirected graph In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called '' ve ...
, and its special case the
bipartite double cover In graph theory, the bipartite double cover of an undirected graph is a bipartite, covering graph of , with twice as many vertices as . It can be constructed as the tensor product of graphs, . It is also called the Kronecker double cover, cano ...
*
Covering group In mathematics, a covering group of a topological group ''H'' is a covering space ''G'' of ''H'' such that ''G'' is a topological group and the covering map is a continuous group homomorphism. The map ''p'' is called the covering homomorphism. ...
*
Galois connection In mathematics, especially in order theory, a Galois connection is a particular correspondence (typically) between two partially ordered sets (posets). Galois connections find applications in various mathematical theories. They generalize the funda ...
*
Quotient space (topology) In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient t ...


Literature

* Allen Hatcher: ''Algebraic Topology''. Cambridge Univ. Press, Cambridge, ISBN 0-521-79160-X * Otto Forster: ''Lectures on Riemann surfaces''. Springer Berlin, München 1991, ISBN 978-3-540-90617-9 * James Munkres: ''Topology''. Upper Saddle River, NJ: Prentice Hall, Inc., ©2000, ISBN 978-0-13-468951-7 * Wolfgang Kühnel: ''Matrizen und Lie-Gruppen''. Springer Fachmedien Wiesbaden GmbH, Stuttgart, ISBN 978-3-8348-9905-7


References

Algebraic topology Homotopy theory Fiber bundles Topological graph theory